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Numerical Relativity and the Discovery of Gravitational Waves Robert A. Eisenstein * MIT LIGO, NW22-272, 185 Albany St., Cambridge, MA 02139 (Dated: April 23, 2018) Solving Einstein’s equations precisely for strong-field gravitational systems is essential to deter- mining the full physics content of gravitational wave detections. Without these solutions it is not possible to extract precise values for initial and final-state system parameters. Obtaining these solutions requires extensive numerical simulations, as Einstein’s equations governing these systems are much too difficult to solve analytically. These difficulties arise principally from the curved, non-linear nature of spacetime in general relativity. Developing the numerical capabilities needed to produce reliable, efficient calculations has required a Herculean 50-year effort involving hundreds of researchers using sophisticated physical insight, algorithm development, computational technique and computers that are a billion times more capable than they were in 1964 when computations were first attempted. My purpose is to give an accessible overview for non-experts of the major developments that have made such dramatic progress possible. I. OVERVIEW OF A BLACK-HOLE BLACK-HOLE COALESCENCE On September 14, 2015, at 09:50:45 UTC the two detectors of the advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) 1 simultane- ously observed 2 the binary black hole merger known as GW150914. The binary pair merged at a luminosity dis- tance of 410 +160 -180 Mpc. Analysis revealed 3 that the two BH masses involved in the coalescence were, in the source frame, 35.8 +5.3 -3.9 and 29.1 +3.8 -4.3 M , while the mass of the final-state BH was 62.0 +4.1 -3.7 M . The difference in mass between the initial and final state, 3.0 +0.5 -0.4 M , was ra- diated away as gravitational radiation. No associated electromagnetic radiation or other cosmic rays were ob- served. Astonishingly, the coalescence and ringdown to a final stable BH took less than 0.2 second (within LIGO’s frequency band), coming after an orbital dance lasting billions of years. This observation, coming 100 years af- ter Einstein’s publication of general relativity, is yet an- other confirmation of its validity. It also is the first direct confirmation that BHs can come in pairs. Figure 1 is a comparison of the observed strains at the Hanford and Livingston LIGO sites after shifting and in- verting the Hanford data to account for the difference in arrival time and the relative orientation of the detectors. The event was identified nearly in real time using detec- tion techniques that made minimal assumptions 4 about the nature of the incoming wave. Subsequent analysis used matched-filter techniques 5 to establish the statisti- cal significance of the observation. Detailed statistical analyses using Bayesian methods were used to estimate the parameters of the coalescing BH–BH system. 3 Long before coalescence occurs, the two orbiting BHs can be represented as point masses co-rotating in an orbit of very large size. This “inspiral” is indicated on the left side of Fig. 2. As the inspiral progresses, the orbit becomes circularized due to energy loss. The spacetime is basically flat except near each BH. Even so, Newtonian physics cannot accurately describe what is happening. FIG. 1. GW strains within a 35–350 Hz passband measured at the Hanford and Livingston LIGO observatories during the detection of GW150914. Time is measured relative to 09:50:45 UTC. The event arrived 6.9 +0.5 -0.4 ms later at Hanford than at Livingston (see text). (From Ref. 2) Instead, “Post-Newtonian” (PN) 6 and “Effective One- Body” (EOB) 7 methods must be employed. As the BH’s near each other (center, Fig. 2), space- time begins to warp significantly and the BH horizons are distorted. The EOB approach provides a good descrip- tion (better than one might expect) until the beginning of coalescence, when the spacetime becomes significantly curved and highly non-linear. In fact, the inspiraling waveform depends strongly on several aspects of the BH– BH interaction, e.g. their masses, spins, polarizations and orbit eccentricity. This dependence plays a key role in the extraction of those parameters, but requires fits to numerical relativity simulations to reproduce the correct result as the binary system approaches merger. Recently, parameter estimation methods have directly used numer- ical relativity simulations 8–10 to do this. Soon after the BH’s reach their “innermost stable cir- cular orbit” (ISCO) they “plunge” together, coalescing into a single highly vibrating, spinning (Kerr) 11 BH. Nu- merical relativity is needed to describe this. The final BH rings down via the emission of gravitational radiation to arXiv:1804.07415v1 [gr-qc] 20 Apr 2018
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Numerical Relativity and the Discovery of Gravitational Waves

Robert A. Eisenstein∗

MIT LIGO, NW22-272, 185 Albany St., Cambridge, MA 02139(Dated: April 23, 2018)

Solving Einstein’s equations precisely for strong-field gravitational systems is essential to deter-mining the full physics content of gravitational wave detections. Without these solutions it is notpossible to extract precise values for initial and final-state system parameters. Obtaining thesesolutions requires extensive numerical simulations, as Einstein’s equations governing these systemsare much too difficult to solve analytically. These difficulties arise principally from the curved,non-linear nature of spacetime in general relativity. Developing the numerical capabilities neededto produce reliable, efficient calculations has required a Herculean 50-year effort involving hundredsof researchers using sophisticated physical insight, algorithm development, computational techniqueand computers that are a billion times more capable than they were in 1964 when computationswere first attempted. My purpose is to give an accessible overview for non-experts of the majordevelopments that have made such dramatic progress possible.

I. OVERVIEW OF A BLACK-HOLEBLACK-HOLE COALESCENCE

On September 14, 2015, at 09:50:45 UTC thetwo detectors of the advanced Laser InterferometerGravitational-Wave Observatory (aLIGO)1 simultane-ously observed2 the binary black hole merger known asGW150914. The binary pair merged at a luminosity dis-tance of 410+160

−180 Mpc. Analysis revealed3 that the twoBH masses involved in the coalescence were, in the sourceframe, 35.8+5.3

−3.9 and 29.1+3.8−4.3 M�, while the mass of the

final-state BH was 62.0+4.1−3.7 M�. The difference in mass

between the initial and final state, 3.0+0.5−0.4 M�, was ra-

diated away as gravitational radiation. No associatedelectromagnetic radiation or other cosmic rays were ob-served. Astonishingly, the coalescence and ringdown to afinal stable BH took less than 0.2 second (within LIGO’sfrequency band), coming after an orbital dance lastingbillions of years. This observation, coming 100 years af-ter Einstein’s publication of general relativity, is yet an-other confirmation of its validity. It also is the first directconfirmation that BHs can come in pairs.

Figure 1 is a comparison of the observed strains at theHanford and Livingston LIGO sites after shifting and in-verting the Hanford data to account for the difference inarrival time and the relative orientation of the detectors.The event was identified nearly in real time using detec-tion techniques that made minimal assumptions4 aboutthe nature of the incoming wave. Subsequent analysisused matched-filter techniques5 to establish the statisti-cal significance of the observation. Detailed statisticalanalyses using Bayesian methods were used to estimatethe parameters of the coalescing BH–BH system.3

Long before coalescence occurs, the two orbiting BHscan be represented as point masses co-rotating in an orbitof very large size. This “inspiral” is indicated on theleft side of Fig. 2. As the inspiral progresses, the orbitbecomes circularized due to energy loss. The spacetimeis basically flat except near each BH. Even so, Newtonianphysics cannot accurately describe what is happening.

FIG. 1. GW strains within a 35–350 Hz passband measuredat the Hanford and Livingston LIGO observatories duringthe detection of GW150914. Time is measured relative to09:50:45 UTC. The event arrived 6.9+0.5

−0.4 ms later at Hanfordthan at Livingston (see text). (From Ref. 2)

Instead, “Post-Newtonian” (PN)6 and “Effective One-Body” (EOB)7 methods must be employed.

As the BH’s near each other (center, Fig. 2), space-time begins to warp significantly and the BH horizons aredistorted. The EOB approach provides a good descrip-tion (better than one might expect) until the beginningof coalescence, when the spacetime becomes significantlycurved and highly non-linear. In fact, the inspiralingwaveform depends strongly on several aspects of the BH–BH interaction, e.g. their masses, spins, polarizations andorbit eccentricity. This dependence plays a key role inthe extraction of those parameters, but requires fits tonumerical relativity simulations to reproduce the correctresult as the binary system approaches merger. Recently,parameter estimation methods have directly used numer-ical relativity simulations8–10 to do this.

Soon after the BH’s reach their “innermost stable cir-cular orbit” (ISCO) they “plunge” together, coalescinginto a single highly vibrating, spinning (Kerr)11 BH. Nu-merical relativity is needed to describe this. The final BHrings down via the emission of gravitational radiation to

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FIG. 2. Top: A schematic drawing of the inspiral, plunge,merger and ringdown of two coalescing BHs (see text). Bot-tom: Comparison of a best-fit template of the measured straindata to the predicted unfiltered theoretical waveform, calcu-lated using the extracted physical parameters. (From Ref. 2)

a stable, spinning, non-radiating BH. The ringdown canbe described using a perturbative quasi-normal modesmodel.12 An overview of the basic physics of the entireBH–BH merger is available in Ref. 13.

II. EINSTEIN’S EQUATIONS

Einstein’s equations,14,15 written in final form inNovember, 1915, are expressed in terms of the four gen-eralized coordinates of spacetime, which is representedas a geometrical Riemann manifold 16 M that extendsto infinity in all directions.17 Three of the coordinates(labeled 1-3) are spatial and one (labeled 0) representstime. At this stage, they are not represented by a specificcoordinate system. The manifold shape is determined bythe real 4-by-4 metric tensor gµν , which in Einstein’s the-ory is determined by the mass densities and energy fluxespresent at every point in spacetime. These relationshipsare summarized by Einstein’s equations written in tensorform:18,19

Gµν := Rµν −1

2gµνR = 8πTµν (1)

The quantity Gµν , Einstein’s tensor,20 is defined in termsof the metric tensor gµν , the Ricci curvature tensor21

Rµν and the Ricci scalar22 R. The energy-momentum, orstress-energy, tensor is represented by Tµν .

A remarkable feature of Einstein’s equations is that thegeometry of spacetime appears only on the left-hand side,imbedded in Gµν , while the physical momentum-energy

content appears only on the right, imbedded in Tµν .Thus, as John Wheeler memorably remarked: “Mattertells spacetime how to curve, and spacetime tells matterhow to move.”

The metric tensor gµν plays the same role in gen-eral relativity as it does in special relativity. In eachcase it provides the link between the generalized coordi-nates xµ and the invariant spacetime interval ds: ds2 =gµνdx

µdxν , summing as usual over repeated indices. Inspecial relativity it defines a flat (Minkowski)23 space.In general relativity it defines the curved (Riemannian)16

manifoldM. The curvature, due to gravitational sources,enters via the Ricci tensor Rµν and the Ricci scalar R.Thus in both special and general relativity, the metrictensor elements determine all the physical observableswe can calculate.

The subscripts (µ, ν) range over the integers 0 to 3,implying the need to solve a system of 16 coupled equa-tions. However, the symmetries of the metric limit theactual number to 10. The simple appearance of Ein-stein’s equations in tensor notation masks a very greatdeal of complexity. When written out in full they cancontain thousands of terms. These will have significantnon-linearities due to the spacetime curvature that oc-curs when the gravitational fields are very strong.

III. SOLVING EINSTEIN’S EQUATIONS

Due to the complexities mentioned above, there arevery few analytical solutions of Einstein’s equations ofphysical relevance. The ones we know of arise in situ-ations involving a high degree of symmetry. Most im-portant for the present discussion are the Schwarzschildsolution24 (for a spherically-symmetric mass M with spin0) and the Kerr solution11 (for a spherically-symmetricmass M with spin J). Exact solutions that include acharge Q on the BH (an unlikely prospect) have also beenfound but will not be discussed here.

Schwarzschild’s 1916 discovery led to one of the mostimportant predictions of general relativity: the existenceof BH’s. A valuable simplification comes in the formof the “no-hair” conjecture,25 which states that in fourdimensions the BH solutions to Einstein’s equations canonly depend on the mass, spin and charge of the BH.

Einstein predicted the existence of gravitationalwaves26–28 moving at the speed of light29 in 1916. Rea-soning by analogy to electromagnetism (i.e. acceleratingmasses should radiate gravitational waves as acceleratingcharges radiate electromagnetic ones),30 he found themby linearizing Einstein equations for the case of nearly flatspacetime (i.e. weak gravitation). For many years therewas considerable uncertainty as to their existence, evenfrom Einstein himself, but the issue was put to rest31 inthe mid-1950’s. Gravitational waves exist in the strongfield case also, but the equations describing them are notlinear. It is those equations we must solve numericallyin order to observe and quantify the nature of BH–BH,

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BH–Neutron Star (NS) or NS–NS coalescences.As if BHs and gravitational waves were not enough,

Einstein’s equations also predict that the structure of theUniverse is not static: as time goes on, it will either ex-pand or contract. Since there was no evidence in 1916 foreither of these prospects, Einstein introduced a “cosmo-logical constant” to force his equations to predict a staticUniverse. When the expansion of the Universe32 was es-tablished in 1926, he later called this decision “my great-est blunder.” Ironically, with the discovery33 in 1998 thatthe Universe is accelerating as it expands, the cosmologi-cal constant plays an important role in accounting for (ifnot understanding) the cosmic acceleration.

IV. NUMERICAL RELATIVITY AND BH–BHCOALESCENCE34–38

It is worth pointing out that even though these cal-culations are prodigiously difficult, the BH–BH system isvery likely the simplest strongly–interacting gravitationalproblem we will ever encounter. If the study of strong-field general relativity is to have a future, it is imperativeto solve it.

The long road to stable, convergent numerical solutionsbegan in 1952, when Yvonne Foures-Brouhat39 showedthat Einstein’s equations were well-posed. Simply put,this means: (1) that solutions of the equations exist; and(2) that small changes to initial conditions produce onlystable, continuous (i.e. non-chaotic) changes in the out-put. Given the difficulty of Einstein’s equations, theseseemingly reasonable expectations are far from obvious.

A. The ADM Procedure

During the next several decades, many substantialdifficulties40 had to be overcome to obtain stable, ac-curate solutions. The first was to recast Einstein’s equa-tions in the form of a computable, time-step iterationprocess (i.e. an initial value problem) that would evolvefrom initial conditions (i.e. an initial spacetime), throughBH–BH coalescence, to the boundary conditions for thefinal state. In the world of partial differential equations(PDE’s) this is called a Cauchy problem. In general rel-ativity, this recipe is referred to as a “3+1” approachbecause space and time are separated. This formulationcomes at a price: giving up overall covariance. It wasfirst proposed by Arnowitt, Deser and Misner41 (ADM)in 1962.

In 1979, York rewrote42 the original ADM prescriptionto emphasize its role in evolving the Einstein equations43

rather than as a basis for a theory of quantum gravity(the original intent of the ADM work). His treatmentis now ubiquitously referred to as the “ADM” prescrip-tion. It has spawned many close cousins, all of which arereferred to as “3+1” algorithms (see Sec. IV B).

The basic ADM idea is to decompose the spacetime bycreating a stack of 3-dimensional, spacelike “foliations”,

or slices, each characterized by a fixed coordinate time(see Fig. 3). These we label Σt. The system evolves bymoving with time from one foliation to the next. Theinvariant spacetime interval, formerly written as ds2 =gµνdx

µdxν , becomes in the “3+1” description:

ds2 = (−α2 + βiβi)dt2 + 2βidtdxi + γijdx

idxj (2)

Here the γij are the 3-dimensional metric tensors forthese surfaces. The indices i and j run from 1 to 3.Note that time appears explicitly. The quantity α (the

lapse) and the three βi (the shift vector ~β) are gauge vari-ables44 that may be freely specified but must be chosenwith care. The lapse determines the rate at which oneprogresses perpendicularly from one slice to the next; itcan be varied as the problem evolves. A lot of thoughtgoes into choosing α because it determines the distancebetween the foliations; good choices avoid singularities.The shift vector basically measures how much the spatialcoordinates change between foliations.

Because the foliations Σt are embedded in the overallspacetime manifoldM, they are characterized by the realExtrinsic Curvature Tensor Kij that describes the natureof the embedding:45

Kij =1

2α(∂tγij −Diβj −Djβi) (3)

Here the symbol ∂t is an ordinary partial derivative withrespect to time, and Di is a spatial covariant derivative.

Evolving Einstein’s equations using a “3+1” methodrequires that initial values of gµν and Kij , 12 numbers inall, must be fixed. But they cannot be chosen arbitrarilybecause of the constraints discussed in the next Section.

FIG. 3. A schematic 3+1 ADM decomposition. Σt1 and Σt2

are spacelike 3-dimensional foliations separated by coordinatetime t2 − t1. The quantity αdt, with α the lapse, is the timestep between Σt2 and Σt1. The shift βi measures the changein coordinate xi in moving from the earlier foliation.

Despite the promise of the ADM method, evolving aBH–BH system through coalescence remained elusive.The reason was that its equations are only weakly hy-perbolic (see Sec. IV B and Ref. 46) and so are ill-posed.

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B. ADM Evolution and Constraints

As mentioned in Sec. II, the symmetries of the metrictensor reduce Einstein’s set of 16 equations for the gµνto 10 coupled, non-linear PDE’s.

Evolution equations. Of the 10, six contain spaceand time derivatives up to second order. These equa-tions provide the evolution of the initial spacetime. Theycontain mixtures of hyperbolic and parabolic (i.e. time-dependent) behavior. Hyperbolic equations are basicallywave equations that describe wave propagation at finitespeed. Solutions to wave equations are generally verystable and converge rapidly. Real parabolic equations(e.g. the heat equation) do not exhibit wave-like behav-ior. On the other hand, a parabolic equation with animaginary component (e.g. the Schrodinger equation),exhibits both a wave speed and dispersion.47

Constraint equations. The remaining four equa-tions have no time derivatives and serve as constraintson the time development. They are elliptic (i.e. time-independent) equations. These are often used to describetime-independent boundary-value problems. Because ofthe non-linearity of strong-field general relativity, theyare harder than usual to solve numerically.

In theory, once the constraints are satisfied initiallythey should remain so. But for numerical solutions thatis often not the case, especially when significant non-linearities are present. Small numerical errors can expo-nentially grow. Keeping the constraints satisfied at alltimes has proven essential to reaching stable, convergentsolutions of the BH–BH coalescence problem.

An instructive parallel appears with Maxwell’s equa-tions. There, the laws of Ampere and Faraday, both con-taining time derivatives of the electric and magnetic fieldsE and B, are the evolution equations, while Gauss’s Lawsfor E and B serve as constraints. Since these equationsare linear the constraints are usually well-behaved. Whenthey aren’t, the results are not solutions to Maxwell’sequations. The analogue is true in numerical relativity.

For most rapid convergence the evolution equationsshould be as wave-like (hyperbolic) as possible. Gaugefreedom is useful for this purpose, keeping in mind thatpoor gauge choices can adversely affect well-posedness.The constraint equations have proven very useful here.Since they can always be written in the form C(x, y, z) =0 (e.g. ∇ ·E – 4πρ = 0), one can add them (or multiplesof them) to the evolution equations wherever that mightbe useful. Picking coordinates (a gauge choice) is alsocrucially important.

There are many other ways48,49 to use gauge free-dom to control problems arising from convergence issues,physical or coordinate singularities, numerical round-offerror, and issues associated with boundary problems atBH horizons (among others). Perhaps the most impor-tant lesson in the development of numerical relativity isthat gauge choices (including the choice of coordinates50)are every bit as important as computing power.

Especially important is the 1987 work of Nakamura,

Oohara and Kajima, which presented51 a version of ADMthat showed much better stability. Later, Shibata andNakamura52 (1995) and Baumgarte and Shapiro53 (1998)confirmed and extended those results. These efforts arecommonly known as the BSSNOK approach.54 It was es-sential to achieving full 3-dimensional simulations of BH–BH coalescences and is in wide use today. It confirms theimportance of selecting carefully the best formulation ofEinstein’s equations for the problem at hand.

C. Harmonic Coordinates and Constraint Damping

Beginning with Einstein, harmonic coordinates haveplayed a major role in general relativity.55 As noted inSec. IV, they were used by Foures-Brouhat39 to show thewell-posedness of Einstein’s vacuum equations. Today,in a generalized form,55,56 they are important in solv-ing numerically the BBH coalescence problem.57 Theywork well because they convert Einstein’s equationsinto second-order strongly hyperbolic form. The ADMformulation, with redefinitions of the lapse and shift,can accommodate them as well. The same cautionsabout constraint damping apply. We refer to this over-all approach,57 Generalized Harmonics with ConstraintDamping, as GHCD.

D. Initial Conditions

In the BSSNOK approach, the initial data consist ofvalues for the γij metric and the extrinsic curvature ma-trix Kij . These depend on the initial parameters ofthe BH’s or NS’s, and on the gauge variables α and~β. As mentioned earlier, these cannot all be chosen in-dependently because of the constraint equations, whichin addition are hard to solve numerically. This difficultproblem has been studied extensively.58–61 In the GCHDtreatment, initial values for the four spacetime coordi-nates and their first derivatives are specified. In bothapproaches the initial constraints are imposed and thenenforced throughout the calculation.

E. Excisions and Moving Punctures

We are dealing with simple Schwarzschild or Kerr BHshaving event horizons behind which the singularites arehidden from view (an idea known as cosmic censor-ship62). It led William Unruh63 to suggest in 1984 thatBH singularities could be excised from the calculationso that their influence is never felt outside the horizon.Thus information can flow into, but not out of, a BH.Fig. 4 shows the imminent coalescence of two non-equalBH’s viewed from this perspective. Note the numericalboundary just below the BH horizon.

However, excision comes at the expense of very de-manding boundary conditions. The BH horizons are con-tinuously moving and spurious numerical artifacts can

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arise (including the unphysical emission of gravitationalradiation), making fine-tuning the calculation and en-forcement of the constraints a continuous necessity.

FIG. 4. A body-shaped, two-center coordinate system forunequal mass BHs. “H” labels a BH horizon while “NB” is itsnumerical boundary. No mesh is needed beneath that surface.“CAH” is the common apparent horizon. At far distances thecoordinate lines are close to spherically symmetric. (From M.Scheel, used with permission)

Another approach is to view BHs as Einstein-Rosenbridges64 (See Fig. 5). This was done first by Hahn andLindquist65 in their seminal 1964 calculations of BH–BHcoalescence that founded numerical relativity.66 The sin-gularity lies on the wormhole axis perpendicular to thespacetimes that are above and below. Note that the co-ordinate lines can approach the singularity but cannotreach it. In further developments the wormholes werecompactified into punctures in the spacetime manifold,and then finally into moving punctures67,68 and trum-pets69,70 that could be identified71 as moving BHs. Thecalculations are done so that the grid points avoid punc-ture singularities. Allowing the punctures to move wasthe key step in making this method work.

FIG. 5. Wormhole(left) and trumpet (right) representationof a BH. (From Refs. 69 and 70).

F. Meshes, Coordinates, Numerical Integration

The spatial extension of a BH–BH coalescence is huge.At the beginning, the BHs are widely separated andspacetime is essentially flat except near the BH horizons.Post-Newtonian physics holds sway. Just before coales-cence, the BH’s are only tens to hundreds of kilometers

apart, spacetime is highly curved and general relativity isdominant. Clearly, solving this problem involves wildlydifferent length scales as it moves toward coalescence,with corresponding changes required in the numericalmeshes. Adaptive Mesh Refinement schemes72 have beendeveloped to handle this issue.

The same consideration applies to the choice of coordi-nate system. For a BH–BH system, it is natural to chooseone that has two spherical-polar centers in close, evolvinginto nearly spherical symmetry far away (see Fig. 4). Inaddition, much better numerical accuracy in satisfyingthe boundary conditions at the BH horizons will result ifthe coordinate lines are perpendicular to the BH horizonsurface. We must also account for the motion of the BHsand the distortion of their horizons as the coalescenceevolves. As Fig. 4 shows, this can lead to great numericalcomplexity and the clear need to use curvilinear coordi-nates and non-rectangular mesh schemes.

The numerical integration procedures in most commonuse are finite difference (FD)73 or spectral interpolation(Spec)74–76 methods. Both have long, well-known histo-ries. FD methods yield approximate solutions to PDEsat specific points on the mesh. Spectral methods utilizesmooth functions fitted to several mesh points that canprovide highly accurate values at any location.

G. Numerical calculations of BH–BH coalescence

The pioneering Hahn–Lindquist computation treatedtwo equal-mass BHs that were represented by a manifoldcontaining co-joined wormholes described by Einstein–Rosen bridges. A dozen years later, Smarr andcollaborators77 used a similar model to study the head-oncollision of non-rotating BH’s with emission of gravita-tional radiation.

While neither of these calculations converged to a fi-nite result, at the time there appeared to be no funda-mental obstacle to achieving realistic results once enoughcomputational power could be brought to bear. The sta-bility issues mentioned in Sec. IV B, especially regardinghyperbolicity, maintaining constraints, and how best tohandle the physical BH singularities were not yet fully ap-preciated. Dealing with these issues awaited the arrivalof BSSNOK (ca. 1998), GCHD (2005) and the “movingpunctures” (2005) algorithms.

In 2005, great breakthroughs were achieved byPretorius57 and the Brownsville67 and Goddard68

groups. Working independently and using quite differ-ent methods, they performed stable, accurate simula-tions of BH–BH coalescence that agreed very well witheach other. 78 Fig. 6 compares their calculated polariza-tions for a head-on collision of equal-mass BHs result-ing in the formation of a Kerr BH. Pretorius57 used theGHCD formulation and BH excision. The Brownsville67

and Goddard68 groups used the BSSNOK formulationwith the BHs represented by moving punctures. Theseearly calculations all employed FD integration methods.

It’s not possible to overstate the importance of these

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results. With reliable, highly accurate numerical meth-ods in hand, not only is the full scientific content of thegravity-wave detections revealed, but more realistic cal-culations are possible (e.g. including unequal BH masses,spin effects and eccentric orbits). Detailed calculationsof more complicated gravitational systems, such as NSbinaries79 or NS–BH systems, as well as detailed tests ofstrong-field general relativity,80 have begun. Many codesare available; most use a BSSNOK+FD framework, theothers a GHCD+Spec treatment.81

FIG. 6. Comparison of calculations from Pretorius57 (red),Campanelli et al.67 (blue) and Centrella et al.68 (black). Theabscissa shows time (in units of the final BH mass) and theordinate is the + polarization of the outgoing gravitationalradiation. (From Ref. 78)

H. Inspiral – Merger – Ringdown (IMR) models

To identify possible BH–BH mergers and obtain es-timates of their physical parameters, the data analysesuse “template banks” of strain waveforms that can bematched in real time with incoming strain data.

However, assembling a template bank is a major chal-lenge. Because templates can depend on as many as 17parameters, thousands to millions of them are needed.Since each fully-NR calculation takes weeks to monthsto do, this is a totally impractical goal. Existing fullyrelativistic waveform catalogs82 contain at most a fewthousand templates.

Instead, since most of the strain waveform (inspi-ral to just before merger and the ringdown afterwards)can be accurately modeled using highly efficient post-Newtonian,6 EOB7 and quasi-normal mode12 methods(i.e. without the numerical relativity portion), we cannormalize the EOB or PN parts against existing numeri-cal relativity catalogs to obtain a robust, highly efficientsurrogate for the full calculation. These approaches83–85

are called IMR models and are in wide use. For parame-ter extraction this works very well because many of them(e.g. BH spins, polarizations and orbit eccentricity) arelargely determined by the inspiral part of the waveform,before numerical relativity is necessary. This would notbe true for tests of strong-field general relativity.

I. You can try this at home

Should you wish to do calculations on your own, thereare very helpful resources available: consult the Simulat-ing Extreme Spacetimes (SXS),86 Einstein Toolkit87 andSuper Efficient Numerical Relativity (SENR)76,88 web-sites for more information. Refs. 35–37 also offer numeri-cal examples. The LIGO Open Science Center89 providesdata from gravitational-wave observations along with ac-cess to tutorials and software tools. You can also par-ticipate in the LIGO search for gravitational waves bysigning up with [email protected]

V. FINAL COMMENT

GW150914 was a supernova in the history of physicsand cosmology. It, and the LIGO/VIRGO discover-ies since then, have amazed even the most optimisticamong us. GW170817, the first-ever sighting of a NS–NSmerger and its subsequent electromagnetic counterparts,has provided a remarkable glimpse of the power of multi-messenger astronomy. The last three years have revealedjust how much the “gravitational Universe” has to teachus now that we can see it.

It has taken 100 years to reach this point. Because ofthe genius of Albert Einstein, who saw that the geometryof the Universe was more subtle than realized by IsaacNewton, and the incredible ingenuity of the engineersand scientists of the gravitational science community, wecan now use gravitational waves as a tool to decode theUniverse. But without the generosity and patience of ourfellow citizen-scientists the world over, these discoverieswould not have been possible.

ACKNOWLEDGMENTS

I thank my colleagues at MIT LIGO for many conver-sations about gravitation and cosmology. I also thankThomas Baumgarte, Manuela Campanelli, Mark Han-nam, Erik Katsavounidis, Harald Pfeiffer, Mark Scheel,Frank Tabakin and Rai Weiss for very useful contribu-tions to this manuscript.

[email protected] D. Martynov, et al. “The Sensitivity of the Advanced

LIGO Detectors at the Beginning of Gravitational Wave

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and Virgo Collaboration), “Observation of Gravita-

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tional Waves from a Binary Black Hole Merger”, Phys.Rev. Lett. 116, 061102 (2016), doi: 10.1103/Phys-RevLett.116.061102.

3 B.P. Abbott, et al. (LIGO Scientific Collaboration andVirgo Collaboration), “Properties of the Binary Black HoleMerger GW150914”, arXiv: 1602.03840 [gr-qc].

4 B. P. Abbott, et al. (LIGO Scientific Collaboration andVirgo Collaboration), “Observing gravitational-wave tran-sient GW150914 with minimal assumptions”, Phys. Rev.D93, 122004 (2016).

5 B.P. Abbott, et al. (LIGO Scientific Collaboration andVirgo Collaboration), “GW150914: First results from thesearch for binary black hole coalescence with AdvancedLIGO”, Phys. Rev. D93, 122003 (2016).

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15 J. B. Hartle, Gravity: An Introduction to Einstein’s Gen-eral Relativity, 3rd ed. (Addison-Wesley, San Francisco,2003).

16 A Riemann manifold is a curved space which is locally flatnear each spacetime point. The Riemann curvature tensordescribes the curvature by measuring the change of a vectoras it is transported around a closed path on a manifold,while always remaining parallel to its original orientation.This is referred to as “parallel transport.”

17 For a brief overview see M. R. Dennis, “Tensors and Man-ifolds”, in The Princeton Companion to Applied Mathe-matics, edited by N. J. Higham et al., (Princeton U.P.,Princeton, 2015) pp. 127–130.

18 For a brief overview see: M. A. H. MacCallum, “Einstein’sField Equations”, in Ref. 17, pp. 144–146.

19 Einstein’s equations are often written using units in whichthe speed of light (c) and Newton’s gravitational constant(G) are set equal to 1. Thus 1 M� is equivalent to ∼1.5kmor to ∼ 5µs.

20 The Einstein tensor measures the curvature of the manifoldin a region near each point.

21 The Ricci tensor provides a measure of the difference ingeometry between a given Riemann metric and ordinary

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sic geometry of a Riemann manifold near a given point.23 Minkowski space is described by a flat 4-dimensional man-

ifold in which the time coordinate is treated differentlythan the three space coordinates. Thus Minkowski space,though flat, is not a 4-dimensional Euclidean space.

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27 J. D. E. Creighton and W. G. Anderson, Gravitational-Wave Physics and Astronomy, Wiley Series in Cosmology(Wiley-VCH, Weinheim, 2011), pp. 135-6.

28 Gravitational waves are ripples in spacetime itself ratherthan a disturbance superimposed on it (e.g. emission of anelectromagnetic wave from a vibrating charge). Since thereis no physical mechanism to absorb them, gravitationalwaves can travel cosmological distances at speed c withoutdispersion. This has been confirmed recently in Ref. 29 toabout 1 part in 1015.

29 B. P. Abbott, et al. (LIGO Scientific Collaboration andVirgo Collaboration, Fermi Gamma-ray Burst Monitor,and INTEGRAL), “Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger:GW170817 andGRB 170817A”, ApJ. Lett. 838:L13, 1-27 (2018).

30 An essential difference is that the lowest order of electro-magnetic radiation is the dipole term, while for gravita-tional radiation it is the quadrupole. So any source of grav-itational waves must possess mass distributions with time-varying quadrupole and/or higher multipole moments.

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40 See Ref. 38, Section II-B, and also V. Cardoso, L. Gualtieri,C. Herdeiro and U. Sperhake, “Exploring New PhysicsFrontiers Through Numerical Relativity”, Living Rev. Rel-ativity 18, 1-156 (2015) pp. 9-14.

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46 D. Hilditch, “ An Introduction to Well-Posedness and Free-Evolution”, arXiv:1308.2012v1 [gr-qc].

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86 Simulating Extreme Spacetimes website:<https://www.black-holes.org>.

87 Einstein Toolkit website: <https://einsteintoolkit.org>.88 Super Efficient Numerical Relativity (SENR) website:<https://math.wvu.edu/ zetienne/SENR/index.html>.Also see Ref. 76.

89 https://losc.ligo.org/about/90 Einstein@Home website: <https://www.einsteinathome.org>.


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